Last week, I gave my students the following task, which I first heard about from an instructor at the University Lab School in Honolulu.
Construct a regular hexagon with segment AB as one of its sides.
-You may not use any Shapes tools.
-You many not use any Measurement tools.
-When you are finished, we can use Measurement tools to justify your construction.
How would you construct a regular hexagon without knowing the formal steps for the construction of hexagon with a compass and straightedge?
We were finishing up a unit on Rigid Motions, so the hope was that the students would think about the rigid motions that would map one side of a regular hexagon to another side of a regular hexagon.
A few students asked about the measures of angles in the hexagon. Instead of answering directly, I drew two auxiliary objects to see if that would help them determine the measure of an interior angle of a regular hexagon.
It helped. I didn’t have to tell anyone the angle measure.
While students were working, I monitored their progress using the Class Capture feature of TI-Nspire Navigator. I selected which hexagons we should discuss as a whole class. I sequenced the order in which I called on students.
Most students started with the rotation of a segment. And by far the majority of the students did the entire construction using rotations. Some students were surprised when they rotated segment AB 120° about point B. Segment BA’ appeared below segment AB. Why did that happen?
Some students rotated segment AB 120° about point A. Segment AB’ appeared above segment AB. Why did that happen?
Some students realized that they could use the same degree measure for each of they rotations.
Some students decided to hide the degree measure when they finished to make their construction “look better”.
The big deal was that we were able to grab and move point A or point B at the end of the construction and preserve the hexagon.
A few students made use of the symmetry of the hexagon. P.R. rotated segment AB -120° about point B to get BA’. He rotated segment AB 120° about point A to get AB’. Then he connected A’B’ to create one of the lines of symmetry of the hexagon. He reflected all three segments about line A’B’ to get the rest of the hexagon.
H.J. rotated segment AB -120° about point B to get BA’. Then she found constructed a perpendicular through the midpoint of segment AB. She reflected segment BA’ about the perpendicular bisector of segment AB to get segment AA’’. Then she constructed a line parallel to line AB through A’. She reflected segments AB, BA’ and AA’’ about that parallel line to get the rest of her hexagon.
I am glad for students who see things differently than most of us. And I am even more glad that they are willing to share what they see with us.
And so the journey continues ….